Integrand size = 23, antiderivative size = 429 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=-\frac {b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{40 a d}-\frac {b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{120 a^2 d}+\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{24 a^2 d}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}+\frac {\left (176 a^2-167 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{40 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {a \left (96 a^2+179 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{40 d \sqrt {a+b \sin (c+d x)}}-\frac {5 b \left (12 a^2-b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{8 d \sqrt {a+b \sin (c+d x)}} \] Output:
-1/40*b*(96*a^2-25*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(1/2)/a/d-1/120*b*(208 *a^2-25*b^2)*cos(d*x+c)*(a+b*sin(d*x+c))^(3/2)/a^2/d+1/24*(32*a^2-3*b^2)*c ot(d*x+c)*(a+b*sin(d*x+c))^(5/2)/a^2/d-1/12*b*cot(d*x+c)*csc(d*x+c)*(a+b*s in(d*x+c))^(7/2)/a^2/d-1/3*cot(d*x+c)*csc(d*x+c)^2*(a+b*sin(d*x+c))^(7/2)/ a/d-1/40*(176*a^2-167*b^2)*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)/d/((a+b*sin(d*x+c))/(a+b))^(1/2)-1/40 *a*(96*a^2+179*b^2)*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)/d/(a+b*sin(d*x+c))^(1/2)+5/8*b*(12* a^2-b^2)*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)/d/(a+b*sin(d*x+c))^(1/2)
Result contains complex when optimal does not.
Time = 4.11 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.09 \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\frac {\frac {2 i \left (-176 a^2+167 b^2\right ) \left (-2 a (a-b) E\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (-2 a \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )+b \operatorname {EllipticPi}\left (\frac {a+b}{a},i \text {arcsinh}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right ),\frac {a+b}{a-b}\right )\right )\right ) \sec (c+d x) \sqrt {-\frac {b (-1+\sin (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sin (c+d x))}{a-b}}}{a b \sqrt {-\frac {1}{a+b}}}-\frac {8 a \left (40 a^2-173 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}+\frac {2 b \left (424 a^2+117 b^2\right ) \operatorname {EllipticPi}\left (2,\frac {1}{4} (-2 c+\pi -2 d x),\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{\sqrt {a+b \sin (c+d x)}}-\frac {4}{3} \sqrt {a+b \sin (c+d x)} \left (176 a b \cos (c+d x)+5 \cot (c+d x) \left (-32 a^2+33 b^2+26 a b \csc (c+d x)+8 a^2 \csc ^2(c+d x)\right )+24 b^2 \sin (2 (c+d x))\right )}{160 d} \] Input:
Integrate[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]
Output:
(((2*I)*(-176*a^2 + 167*b^2)*(-2*a*(a - b)*EllipticE[I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*(-2*a*EllipticF[I *ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)] + b*EllipticPi[(a + b)/a, I*ArcSinh[Sqrt[-(a + b)^(-1)]*Sqrt[a + b*Sin[c + d*x]]], (a + b)/(a - b)]))*Sec[c + d*x]*Sqrt[-((b*(-1 + Sin[c + d*x]))/(a + b))]*Sqrt[-((b*(1 + Sin[c + d*x]))/(a - b))])/(a*b*Sqrt[-(a + b)^(-1)]) - (8*a*(40*a^2 - 173*b^2)*EllipticF[(-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]* Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] + (2*b*(424*a ^2 + 117*b^2)*EllipticPi[2, (-2*c + Pi - 2*d*x)/4, (2*b)/(a + b)]*Sqrt[(a + b*Sin[c + d*x])/(a + b)])/Sqrt[a + b*Sin[c + d*x]] - (4*Sqrt[a + b*Sin[c + d*x]]*(176*a*b*Cos[c + d*x] + 5*Cot[c + d*x]*(-32*a^2 + 33*b^2 + 26*a*b *Csc[c + d*x] + 8*a^2*Csc[c + d*x]^2) + 24*b^2*Sin[2*(c + d*x)]))/3)/(160* d)
Time = 3.75 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.04, number of steps used = 27, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.174, Rules used = {3042, 3204, 27, 3042, 3526, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3538, 25, 3042, 3134, 3042, 3132, 3481, 3042, 3142, 3042, 3140, 3286, 3042, 3284}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^{5/2}}{\tan (c+d x)^4}dx\) |
| \(\Big \downarrow \) 3204 |
| \(\displaystyle -\frac {\int \frac {1}{4} \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \left (32 a^2+10 b \sin (c+d x) a-3 b^2-\left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right )dx}{6 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x))^{5/2} \left (32 a^2+10 b \sin (c+d x) a-3 b^2-\left (24 a^2-5 b^2\right ) \sin ^2(c+d x)\right )dx}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {(a+b \sin (c+d x))^{5/2} \left (32 a^2+10 b \sin (c+d x) a-3 b^2-\left (24 a^2-5 b^2\right ) \sin (c+d x)^2\right )}{\sin (c+d x)^2}dx}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle -\frac {\int \frac {1}{2} \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (208 a^2-25 b^2\right ) \sin ^2(c+d x)-6 a \left (8 a^2-5 b^2\right ) \sin (c+d x)+15 b \left (12 a^2-b^2\right )\right )dx-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-b \left (208 a^2-25 b^2\right ) \sin ^2(c+d x)-6 a \left (8 a^2-5 b^2\right ) \sin (c+d x)+15 b \left (12 a^2-b^2\right )\right )dx-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \int \frac {(a+b \sin (c+d x))^{3/2} \left (-b \left (208 a^2-25 b^2\right ) \sin (c+d x)^2-6 a \left (8 a^2-5 b^2\right ) \sin (c+d x)+15 b \left (12 a^2-b^2\right )\right )}{\sin (c+d x)}dx-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {2}{5} \int \frac {3}{2} \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-2 \left (40 a^2-71 b^2\right ) \sin (c+d x) a^2-3 b \left (96 a^2-25 b^2\right ) \sin ^2(c+d x) a+25 b \left (12 a^2-b^2\right ) a\right )dx+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \int \csc (c+d x) \sqrt {a+b \sin (c+d x)} \left (-2 \left (40 a^2-71 b^2\right ) \sin (c+d x) a^2-3 b \left (96 a^2-25 b^2\right ) \sin ^2(c+d x) a+25 b \left (12 a^2-b^2\right ) a\right )dx+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \int \frac {\sqrt {a+b \sin (c+d x)} \left (-2 \left (40 a^2-71 b^2\right ) \sin (c+d x) a^2-3 b \left (96 a^2-25 b^2\right ) \sin (c+d x)^2 a+25 b \left (12 a^2-b^2\right ) a\right )}{\sin (c+d x)}dx+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {2}{3} \int \frac {3 \csc (c+d x) \left (-2 \left (40 a^2-173 b^2\right ) \sin (c+d x) a^3-b \left (176 a^2-167 b^2\right ) \sin ^2(c+d x) a^2+25 b \left (12 a^2-b^2\right ) a^2\right )}{2 \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\int \frac {\csc (c+d x) \left (-2 \left (40 a^2-173 b^2\right ) \sin (c+d x) a^3-b \left (176 a^2-167 b^2\right ) \sin ^2(c+d x) a^2+25 b \left (12 a^2-b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\int \frac {-2 \left (40 a^2-173 b^2\right ) \sin (c+d x) a^3-b \left (176 a^2-167 b^2\right ) \sin (c+d x)^2 a^2+25 b \left (12 a^2-b^2\right ) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3538 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (-\left (a^2 \left (176 a^2-167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )-\frac {\int -\frac {\csc (c+d x) \left (b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (-\left (a^2 \left (176 a^2-167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )+\frac {\int \frac {\csc (c+d x) \left (b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2\right )}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (-\left (a^2 \left (176 a^2-167 b^2\right ) \int \sqrt {a+b \sin (c+d x)}dx\right )+\frac {\int \frac {b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (-\frac {a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (-\frac {a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}dx}{\sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\int \frac {b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {\int \frac {b \left (96 a^2+179 b^2\right ) \sin (c+d x) a^3+25 b^2 \left (12 a^2-b^2\right ) a^2}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3481 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}}dx+a^3 b \left (96 a^2+179 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+a^3 b \left (96 a^2+179 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}}dx}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \int \frac {1}{\sin (c+d x) \sqrt {a+b \sin (c+d x)}}dx+\frac {2 a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3286 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {\csc (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{2} \left (\frac {3}{5} \left (\frac {\frac {25 a^2 b^2 \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \int \frac {1}{\sin (c+d x) \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}}dx}{\sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}+\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\right )+\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
| \(\Big \downarrow \) 3284 |
| \(\displaystyle -\frac {b \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^{7/2}}{12 a^2 d}-\frac {\frac {1}{2} \left (\frac {2 b \left (208 a^2-25 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 d}+\frac {3}{5} \left (\frac {2 a b \left (96 a^2-25 b^2\right ) \cos (c+d x) \sqrt {a+b \sin (c+d x)}}{d}-\frac {2 a^2 \left (176 a^2-167 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\frac {50 a^2 b^2 \left (12 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}+\frac {2 a^3 b \left (96 a^2+179 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),\frac {2 b}{a+b}\right )}{d \sqrt {a+b \sin (c+d x)}}}{b}\right )\right )-\frac {\left (32 a^2-3 b^2\right ) \cot (c+d x) (a+b \sin (c+d x))^{5/2}}{d}}{24 a^2}-\frac {\cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^{7/2}}{3 a d}\) |
Input:
Int[Cot[c + d*x]^4*(a + b*Sin[c + d*x])^(5/2),x]
Output:
-1/12*(b*Cot[c + d*x]*Csc[c + d*x]*(a + b*Sin[c + d*x])^(7/2))/(a^2*d) - ( Cot[c + d*x]*Csc[c + d*x]^2*(a + b*Sin[c + d*x])^(7/2))/(3*a*d) - (-(((32* a^2 - 3*b^2)*Cot[c + d*x]*(a + b*Sin[c + d*x])^(5/2))/d) + ((2*b*(208*a^2 - 25*b^2)*Cos[c + d*x]*(a + b*Sin[c + d*x])^(3/2))/(5*d) + (3*((2*a*b*(96* a^2 - 25*b^2)*Cos[c + d*x]*Sqrt[a + b*Sin[c + d*x]])/d - (2*a^2*(176*a^2 - 167*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^3*b*(96*a^2 + 179*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]]) + (50*a^2*b^2*(12*a^2 - b^2)*Ellipt icPi[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))/5)/2)/(24*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[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> Simp[(-Cos[e + f*x])*((a + b*Sin[e + f*x])^(m + 1)/(3*a*f*Sin[ e + f*x]^3)), x] + (-Simp[b*(m - 2)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(6*a^2*f*Sin[e + f*x]^2)), x] - Simp[1/(6*a^2) Int[((a + b*Sin[e + f* x])^m/Sin[e + f*x]^2)*Simp[8*a^2 - b^2*(m - 1)*(m - 2) + a*b*m*Sin[e + f*x] - (6*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2, x], x], x]) /; FreeQ[{a, b, e, f , m}, x] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1] && IntegerQ[2*m]
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[(((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. \(1525\) vs. \(2(399)=798\).
Time = 8.89 (sec) , antiderivative size = 1526, normalized size of antiderivative = 3.56
Input:
int(cot(d*x+c)^4*(a+b*sin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/120*(240*a^5*((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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/ 2),((a-b)/(a+b))^(1/2))*sin(d*x+c)^3+288*((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)*EllipticF((( a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^4*b*sin(d*x+c)^3-1566* b^2*((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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/( a+b))^(1/2))*a^3*sin(d*x+c)^3+537*b^3*((a+b*sin(d*x+c))/(a-b))^(1/2)*(-(si n(d*x+c)-1)*b/(a+b))^(1/2)*(-b*(1+sin(d*x+c))/(a-b))^(1/2)*EllipticF(((a+b *sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2))*a^2*sin(d*x+c)^3+501*((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)*EllipticF(((a+b*sin(d*x+c))/(a-b))^(1/2),((a-b)/(a+b))^(1/2 ))*a*b^4*sin(d*x+c)^3-528*((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^5*sin(d*x+c)^3+1029*((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^3*b^2* sin(d*x+c)^3-501*((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^4*sin(d*x+c)^3+900*((a+b*sin(d*x+c))/(a-b...
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(5/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)**4*(a+b*sin(d*x+c))**(5/2),x)
Output:
Timed out
\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cot \left (d x + c\right )^{4} \,d x } \] Input:
integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(5/2),x, algorithm="maxima")
Output:
integrate((b*sin(d*x + c) + a)^(5/2)*cot(d*x + c)^4, x)
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\text {Timed out} \] Input:
integrate(cot(d*x+c)^4*(a+b*sin(d*x+c))^(5/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:
int(cot(c + d*x)^4*(a + b*sin(c + d*x))^(5/2),x)
Output:
int(cot(c + d*x)^4*(a + b*sin(c + d*x))^(5/2), x)
\[ \int \cot ^4(c+d x) (a+b \sin (c+d x))^{5/2} \, dx=\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2}d x \right ) b^{2}+2 \left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4} \sin \left (d x +c \right )d x \right ) a b +\left (\int \sqrt {\sin \left (d x +c \right ) b +a}\, \cot \left (d x +c \right )^{4}d x \right ) a^{2} \] Input:
int(cot(d*x+c)^4*(a+b*sin(d*x+c))^(5/2),x)
Output:
int(sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4*sin(c + d*x)**2,x)*b**2 + 2*i nt(sqrt(sin(c + d*x)*b + a)*cot(c + d*x)**4*sin(c + d*x),x)*a*b + int(sqrt (sin(c + d*x)*b + a)*cot(c + d*x)**4,x)*a**2