Integrand size = 29, antiderivative size = 449 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{64 a^2 d}+\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{192 a^2 d}+\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{96 a^2 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}+\frac {b \left (492 a^2-5 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{64 a d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {b \left (148 a^2+169 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (48 a^4-360 a^2 b^2-5 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{64 a d \sqrt {a+b \sin (c+d x)}} \]
5/192*b*(68*a^2+b^2)*cot(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/96*(60*a^2+ b^2)*cot(d*x+c)*csc(d*x+c)*(a+b*sin(d*x+c))^(5/2)/a^2/d+1/24*b*cot(d*x+c)* csc(d*x+c)^2*(a+b*sin(d*x+c))^(7/2)/a^2/d-1/4*cot(d*x+c)*csc(d*x+c)^3*(a+b *sin(d*x+c))^(7/2)/a/d-1/64*b^2*(196*a^2+5*b^2)*cos(d*x+c)*(a+b*sin(d*x+c) )^(1/2)/a^2/d-1/64*b*(492*a^2-5*b^2)*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/s in(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/d/((a+b*sin(d*x+c))/(a+b))^(1/2)+1/64 *b*(148*a^2+169*b^2)*(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)/d/(a+b*sin(d*x+c))^(1/2)-1/64*(48*a^4-360*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)*El lipticPi(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/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 7.33 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.46 \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\left (-\frac {2}{3} b^2 \cos (c+d x)+\frac {5 \left (116 a^2 b \cos (c+d x)-3 b^3 \cos (c+d x)\right ) \csc (c+d x)}{192 a}+\frac {1}{96} \left (60 a^2 \cos (c+d x)-59 b^2 \cos (c+d x)\right ) \csc ^2(c+d x)-\frac {17}{24} a b \cot (c+d x) \csc ^2(c+d x)-\frac {1}{4} a^2 \cot (c+d x) \csc ^3(c+d x)\right ) \sqrt {a+b \sin (c+d x)}}{d}+\frac {-\frac {2 \left (688 a^3 b-348 a b^3\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 \left (96 a^4-228 a^2 b^2-15 b^4\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (-c+\frac {\pi }{2}-d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {2 i \left (-492 a^2 b^2+5 b^4\right ) \cos (c+d x) \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )-b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sqrt {\frac {b-b \sin (c+d x)}{a+b}} \sqrt {-\frac {b+b \sin (c+d x)}{a-b}}}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(c+d x)} \left (-2 a^2+b^2+4 a (a+b \sin (c+d x))-2 (a+b \sin (c+d x))^2\right ) \sqrt {-\frac {a^2-b^2-2 a (a+b \sin (c+d x))+(a+b \sin (c+d x))^2}{b^2}}}}{256 a d} \]
(((-2*b^2*Cos[c + d*x])/3 + (5*(116*a^2*b*Cos[c + d*x] - 3*b^3*Cos[c + d*x ])*Csc[c + d*x])/(192*a) + ((60*a^2*Cos[c + d*x] - 59*b^2*Cos[c + d*x])*Cs c[c + d*x]^2)/96 - (17*a*b*Cot[c + d*x]*Csc[c + d*x]^2)/24 - (a^2*Cot[c + d*x]*Csc[c + d*x]^3)/4)*Sqrt[a + b*Sin[c + d*x]])/d + ((-2*(688*a^3*b - 34 8*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*(96*a^4 - 228*a^2*b^2 - 15* 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)*(-492*a^2*b^2 + 5*b^4)* 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* 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 - 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)]))/ (256*a*d)
Time = 3.65 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.01, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.931, Rules used = {3042, 3372, 27, 3042, 3526, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^4 (a+b \sin (c+d x))^{5/2}}{\sin (c+d x)^5}dx\) |
\(\Big \downarrow \) 3372 |
\(\displaystyle -\frac {\int \frac {1}{4} \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \left (60 a^2+10 b \sin (c+d x) a+b^2-3 \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{12 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^{5/2} \left (60 a^2+10 b \sin (c+d x) a+b^2-3 \left (16 a^2+b^2\right ) \sin ^2(c+d x)\right )dx}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (60 a^2+10 b \sin (c+d x) a+b^2-3 \left (16 a^2+b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^3}dx}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle -\frac {\frac {1}{2} \int \frac {1}{2} \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (-3 b \left (124 a^2+5 b^2\right ) \sin ^2(c+d x)-6 a \left (12 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (68 a^2+b^2\right )\right )dx-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{4} \int \csc ^2(c+d x) (a+b \sin (c+d x))^{3/2} \left (-3 b \left (124 a^2+5 b^2\right ) \sin ^2(c+d x)-6 a \left (12 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (68 a^2+b^2\right )\right )dx-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \int \frac {(a+b \sin (c+d x))^{3/2} \left (-3 b \left (124 a^2+5 b^2\right ) \sin (c+d x)^2-6 a \left (12 a^2-5 b^2\right ) \sin (c+d x)+5 b \left (68 a^2+b^2\right )\right )}{\sin (c+d x)^2}dx-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3526 |
\(\displaystyle -\frac {\frac {1}{4} \left (\int -\frac {3}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (48 a^4-360 b^2 a^2+2 b \left (148 a^2-5 b^2\right ) \sin (c+d x) a-5 b^4+3 b^2 \left (196 a^2+5 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (48 a^4-360 b^2 a^2+2 b \left (148 a^2-5 b^2\right ) \sin (c+d x) a-5 b^4+3 b^2 \left (196 a^2+5 b^2\right ) \sin ^2(c+d x)\right )dx-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \int \frac {\sqrt {a+b \sin (c+d x)} \left (48 a^4-360 b^2 a^2+2 b \left (148 a^2-5 b^2\right ) \sin (c+d x) a-5 b^4+3 b^2 \left (196 a^2+5 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)}dx-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3528 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {2}{3} \int \frac {3 \csc (c+d x) \left (2 b \left (172 a^2-87 b^2\right ) \sin (c+d x) a^2+b^2 \left (492 a^2-5 b^2\right ) \sin ^2(c+d x) a+\left (48 a^4-360 b^2 a^2-5 b^4\right ) a\right )}{2 \sqrt {a+b \sin (c+d x)}}dx-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\int \frac {\csc (c+d x) \left (2 b \left (172 a^2-87 b^2\right ) \sin (c+d x) a^2+b^2 \left (492 a^2-5 b^2\right ) \sin ^2(c+d x) a+\left (48 a^4-360 b^2 a^2-5 b^4\right ) a\right )}{\sqrt {a+b \sin (c+d x)}}dx-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\int \frac {2 b \left (172 a^2-87 b^2\right ) \sin (c+d x) a^2+b^2 \left (492 a^2-5 b^2\right ) \sin (c+d x)^2 a+\left (48 a^4-360 b^2 a^2-5 b^4\right ) a}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3538 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (a b \left (492 a^2-5 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx-\frac {\int -\frac {\csc (c+d x) \left (a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (a b \left (492 a^2-5 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {\csc (c+d x) \left (a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (a b \left (492 a^2-5 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx+\frac {\int \frac {a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3134 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3132 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {\int \frac {a b \left (48 a^4-360 b^2 a^2-5 b^4\right )-a^2 b^2 \left (148 a^2+169 b^2\right ) \sin (c+d x)}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3481 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx-a^2 b^2 \left (148 a^2+169 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-a^2 b^2 \left (148 a^2+169 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3142 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3140 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx-\frac {2 a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3286 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {1}{4} \left (-\frac {3}{2} \left (\frac {\frac {a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}-\frac {2 a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
\(\Big \downarrow \) 3284 |
\(\displaystyle \frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{24 a^2 d}-\frac {\frac {1}{4} \left (-\frac {5 b \left (68 a^2+b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{3/2}}{d}-\frac {3}{2} \left (-\frac {2 b^2 \left (196 a^2+5 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}+\frac {2 a b \left (492 a^2-5 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {2 a b \left (48 a^4-360 a^2 b^2-5 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}-\frac {2 a^2 b^2 \left (148 a^2+169 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )\right )-\frac {\left (60 a^2+b^2\right ) \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2}}{2 d}}{48 a^2}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^{7/2}}{4 a d}\) |
(b*Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(24*a^2*d) - (C ot[c + d*x]*Csc[c + d*x]^3*(a + b*Sin[c + d*x])^(7/2))/(4*a*d) - (-1/2*((6 0*a^2 + b^2)*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/d + ((- 5*b*(68*a^2 + b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/d - (3*((-2*b^ 2*(196*a^2 + 5*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d + (2*a*b*(492 *a^2 - 5*b^2)*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^2*b^2*(148*a^2 + 169*b^2)*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]]) + (2*a*b*(48*a^4 - 360*a^2* b^2 - 5*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)/(48*a^2)
3.12.65.3.1 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[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x ])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^ 2)/(Sqrt[(a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[C/(b*d) Int[Sqrt[a + b*Sin[e + f*x]], x] , x] - Simp[1/(b*d) Int[Simp[a*c*C - A*b*d + (b*c*C - b*B*d + a*C*d)*Sin[ e + f*x], x]/(Sqrt[a + b*Sin[e + f*x]]*(c + d*Sin[e + f*x])), x], x] /; Fre eQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0 ] && NeQ[c^2 - d^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1776\) vs. \(2(510)=1020\).
Time = 21.63 (sec) , antiderivative size = 1777, normalized size of antiderivative = 3.96
1/192*(-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)*EllipticE(((a+b*sin(d*x+c))/(a-b))^(1/2),( (a-b)/(a+b))^(1/2))*a*b^4*sin(d*x+c)^4+444*((a+b*sin(d*x+c))/(a-b))^(1/2)* (-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticF( ((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^4-199 8*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(sin(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin( d*x+c))*b/(a-b))^(1/2)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+ b))^(1/2))*a^3*b^2*sin(d*x+c)^4+507*((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*s in(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*b^3*sin(d*x+c)^4+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^4*sin(d*x+c)^4+144*((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^4*b*sin(d*x+c)^4+1080*((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^2*sin(d*x+c)^4-1080*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(s in(d*x+c)-1)*b/(a+b))^(1/2)*(-(1+sin(d*x+c))*b/(a-b))^(1/2)*EllipticPi(((a +b*sin(d*x+c))/(a-b))^(1/2),(a-b)/a,((a-b)/(a+b))^(1/2))*a^2*b^3*sin(d*...
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
\[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \csc \left (d x + c\right ) \,d x } \]
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Hanged} \]